 Detailed balance without rapid convergence

Detailed balance does not imply that a Markov chain is an irreducible Markov chain: the identity transition matrix is reversible for every probability distribution. Even for an irreducible Markov chain with aperiodicity, the two-state matrix with diagonal $1-\epsilon$ and off-diagonal $\epsilon$, $0<\epsilon <1/2$, has uniform stationary distribution and nontrivial eigenvalue $1-2\epsilon$. From either fixed state its total variation distance after $n$ steps is $\frac{1}{2}(1-2\epsilon {)}^n$, so convergence can be arbitrarily slow. Invariance, reachability and mixing time are distinct questions.